Algorithmic Number Theory

Beschreibung

Beschreibung

The fourth Algorithmic Number Theory Symposium takes place at the U- versiteit Leiden, in the Netherlands, from July 2-7, 2000. Its organization is a joint e?ort of Dutch number theorists from Leiden, Groningen, Nijmegen, and Amsterdam. Sixinvitedtalksand36contributedtalksarescheduled. Thisvolumecontains the written versions of the talks, with the exception of two of the invited talks. Not included are: A rational approach to? by Frits Beukers (Utrecht) and The 40 trillionth binary digit of? is 0 by Peter Borwein (Burnaby, Canada). These talksare aimed at a wider audience, and formpart of thespecial ANTS IV event Pi in de Pieterskerk on July 5, 2000. This event includes an evening ceremony in which the tombstone of Ludolph van Ceulen is replaced. Van Ceulen, who was appointed to Leiden in 1600, calculated 35 decimals of?.Histombstonein the Pieterskerk, in which these decimals were engraved, disappeared in the 19th century. ANTS in Leiden is the fourth in a series of symposia that started in 1994. Previous locations were Cornell University, Ithaca, New York (1994), Universit¿ e de Bordeaux I in Bordeaux, France (1996), and Reed College, Portland, Oregon (1998). The diversity of the papers contained in this volume shows that the maintheme of ANTS, algorithmicnumber theory, istaken ina broadsense. The number of submissions for the Leiden conference largely exceeded the physical limitationsof ourone-week schedule. Weare therefore con?dent thatwe areonly at the beginning of a continuing tradition.

Inhaltsverzeichnis

Invited Talks.- The Complexity of Some Lattice Problems.- Rational Points Near Curves and Small Nonzero | x 3 ? y 2| via Lattice Reduction.- Coverings of Curves of Genus 2.- Lattice Reduction in Cryptology: An Update.- Contributed Papers.- Construction of Secure C ab Curves Using Modular Curves.- Curves over Finite Fields with Many Rational Points Obtained by Ray Class Field Extensions.- New Results on Lattice Basis Reduction in Practice.- Baby-Step Giant-Step Algorithms for Non-uniform Distributions.- On Powers as Sums of Two Cubes.- Factoring Polynomials over ?-Adic Fields.- Strategies in Filtering in the Number Field Sieve.- Factoring Polynomials over Finite Fields and Stable Colorings of Tournaments.- Computing Special Values of Partial Zeta Functions.- Construction of Tables of Quartic Number Fields.- Counting Discriminants of Number Fields of Degree up to Four.- On Reconstruction of Algebraic Numbers.- Dissecting a Sieve to Cut Its Need for Space.- Counting Points on Hyperelliptic Curves over Finite Fields.- Modular Forms for GL(3) and Galois Representations.- Modular Symbols and Hecke Operators.- Fast Jacobian Group Arithmetic on C ab Curves.- Lifting Elliptic Curves and Solving the Elliptic Curve Discrete Logarithm Problem.- A One Round Protocol for Tripartite Diffie-Hellman.- On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields.- Component Groups of Quotients of J 0(N).- Fast Computation of Relative Class Numbers of CM-Fields.- On Probable Prime Testing and the Computation of Square Roots mod n.- Improving Group Law Algorithms for Jacobians of Hyperelliptic Curves.- Central Values of Artin L-Functions for Quaternion Fields.- The Pseudoprimes up to 1013.- Computing the Number of Goldbach Partitions up to 5 108.- Numerical Verification of the Brumer-Stark Conjecture.- Explicit Models of Genus 2 Curves with Split CM.- Reduction in Purely Cubic Function Fields of Unit Rank One.- Factorization in the Composition Algebras.- A Fast Algorithm for Approximately Counting Smooth Numbers.- Computing All Integer Solutions of a General Elliptic Equation.- A Note on Shanks's Chains of Primes.- Asymptotically Fast Discrete Logarithms in Quadratic Number Fields.- Asymptotically Fast GCD Computation in ?[i].